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Thermodynamical considerations implying wall/particles scattering kernels
Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation
1. | School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China, China |
References:
[1] |
A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.
doi: 10.1081/PDE-100002246. |
[2] |
M. Bisi, J. A. Carrillo and G. Toscani, Contractive metrics for a Boltzmann equation for granular gases: Diffusive equilibria, J. Stat. Phys., 118 (2005), 301-331.
doi: 10.1007/s10955-004-8785-5. |
[3] |
C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[4] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. |
[5] |
R. J. DiPerna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation, Comm. Math. Phys., 120 (1988), 1-23.
doi: 10.1007/BF01223204. |
[6] |
R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.
doi: 10.1007/s00220-010-1110-z. |
[7] |
R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[8] |
R. Duan, T. Yang and H. Zhao, The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Differential Equations, 252 (2012), 6356-6386.
doi: 10.1016/j.jde.2012.03.012. |
[9] |
R. Duan, T. Yang and H. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Models Methods Appl. Sci., 23 (2013), 979-1028.
doi: 10.1142/S0218202513500012. |
[10] |
R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[11] |
F. Golse, B. Perthame and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation, Arch. Rational Mech. Anal., 103 (1988), 81-96.
doi: 10.1007/BF00292921. |
[12] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[13] |
Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.
doi: 10.1007/s00205-003-0262-9. |
[14] |
Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[15] |
Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.
doi: 10.1002/cpa.20121. |
[16] |
K. Hamdache, Estimations uniformes des solutions de l'équation de Boltzmann par les méthodes de viscosité artificielle et de diffusion de Fokker-Planck, (French) [Uniform estimates for solutions of the perturbed Boltzmann equation by artificial viscosity or Fokker-Planck diffusion], C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 187-190. |
[17] |
S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph.D thesis, Kyoto University, 1983. Available from: http://hdl.handle.net/2433/97887. |
[18] |
H.-L. Li and A. Matsumura, Behaviour of the Fokker-Planck-Boltzmann equation near a Maxwellian, Arch. Ration. Mech. Anal., 189 (2008), 1-44.
doi: 10.1007/s00205-007-0057-5. |
[19] |
T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Phys. D, 188 (2004), 178-192.
doi: 10.1016/j.physd.2003.07.011. |
[20] |
T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.
doi: 10.1007/s00220-003-1030-2. |
[21] |
S. K. Loyalka, Rarefied gas dynamic problems in environmental sciences, in Proceedings 15th International Symposium on Rarefied Gas Dynamics (eds. V. Boffi and C. Cercignani), Teubner, Stuttgart, 1986. |
[22] |
C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.
doi: 10.1080/03605300600635004. |
[23] |
W. A. Strauss, Decay and asymptotics for $u_{t t} - \Delta u=F(u)$, J. Functional Analysis, 2 (1968), 409-457.
doi: 10.1016/0022-1236(68)90004-9. |
[24] |
R. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
doi: 10.1007/s00205-007-0067-3. |
[25] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, 2002, 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
[26] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[27] |
S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.
doi: 10.3792/pja/1195519027. |
[28] |
S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Anal. Appl. (Singap.), 4 (2006), 263-310.
doi: 10.1142/S0219530506000784. |
[29] |
M.-Y. Zhong and H.-L. Li, Long time behavior of the Fokker-Planck-Boltzmann equation with soft potential, Quart. Appl. Math., 70 (2012), 721-742.
doi: 10.1090/S0033-569X-2012-01269-3. |
show all references
References:
[1] |
A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26 (2001), 43-100.
doi: 10.1081/PDE-100002246. |
[2] |
M. Bisi, J. A. Carrillo and G. Toscani, Contractive metrics for a Boltzmann equation for granular gases: Diffusive equilibria, J. Stat. Phys., 118 (2005), 301-331.
doi: 10.1007/s10955-004-8785-5. |
[3] |
C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[4] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994. |
[5] |
R. J. DiPerna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation, Comm. Math. Phys., 120 (1988), 1-23.
doi: 10.1007/BF01223204. |
[6] |
R. Duan, M. Fornasier and G. Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys., 300 (2010), 95-145.
doi: 10.1007/s00220-010-1110-z. |
[7] |
R. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[8] |
R. Duan, T. Yang and H. Zhao, The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Differential Equations, 252 (2012), 6356-6386.
doi: 10.1016/j.jde.2012.03.012. |
[9] |
R. Duan, T. Yang and H. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Models Methods Appl. Sci., 23 (2013), 979-1028.
doi: 10.1142/S0218202513500012. |
[10] |
R. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[11] |
F. Golse, B. Perthame and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation, Arch. Rational Mech. Anal., 103 (1988), 81-96.
doi: 10.1007/BF00292921. |
[12] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[13] |
Y. Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.
doi: 10.1007/s00205-003-0262-9. |
[14] |
Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
[15] |
Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.
doi: 10.1002/cpa.20121. |
[16] |
K. Hamdache, Estimations uniformes des solutions de l'équation de Boltzmann par les méthodes de viscosité artificielle et de diffusion de Fokker-Planck, (French) [Uniform estimates for solutions of the perturbed Boltzmann equation by artificial viscosity or Fokker-Planck diffusion], C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 187-190. |
[17] |
S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Ph.D thesis, Kyoto University, 1983. Available from: http://hdl.handle.net/2433/97887. |
[18] |
H.-L. Li and A. Matsumura, Behaviour of the Fokker-Planck-Boltzmann equation near a Maxwellian, Arch. Ration. Mech. Anal., 189 (2008), 1-44.
doi: 10.1007/s00205-007-0057-5. |
[19] |
T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Phys. D, 188 (2004), 178-192.
doi: 10.1016/j.physd.2003.07.011. |
[20] |
T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.
doi: 10.1007/s00220-003-1030-2. |
[21] |
S. K. Loyalka, Rarefied gas dynamic problems in environmental sciences, in Proceedings 15th International Symposium on Rarefied Gas Dynamics (eds. V. Boffi and C. Cercignani), Teubner, Stuttgart, 1986. |
[22] |
C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.
doi: 10.1080/03605300600635004. |
[23] |
W. A. Strauss, Decay and asymptotics for $u_{t t} - \Delta u=F(u)$, J. Functional Analysis, 2 (1968), 409-457.
doi: 10.1016/0022-1236(68)90004-9. |
[24] |
R. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
doi: 10.1007/s00205-007-0067-3. |
[25] |
C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, 2002, 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
[26] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[27] |
S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184.
doi: 10.3792/pja/1195519027. |
[28] |
S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Anal. Appl. (Singap.), 4 (2006), 263-310.
doi: 10.1142/S0219530506000784. |
[29] |
M.-Y. Zhong and H.-L. Li, Long time behavior of the Fokker-Planck-Boltzmann equation with soft potential, Quart. Appl. Math., 70 (2012), 721-742.
doi: 10.1090/S0033-569X-2012-01269-3. |
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